FLEXURA E ISOSTASIA DA LITOSFERA

Isostasia

Princípio de Arquimedes

Any floating object displaces its own weight of fluid. — Archimedes of Syracuse

Princípio de Arquimedes

Any floating object displaces its own weight of fluid. — Archimedes of Syracuse

Princípio de Arquimedes

Any floating object displaces its own weight of fluid. — Archimedes of Syracuse

Isostasia da litosfera

Crosta

Manto Litosférico

Astenosfera

Isostasia da litosfera

Crosta

Manto Litosférico

Astenosfera

Isostasia da litosfera

Crosta

Manto Litosférico

AstenosferaProfundidade

de compensação

Isostasia e flexura da litosfera

Isostasia Local

Isostasia Flexural

Isostasia e flexura da litosfera

Isostasia Local

Isostasia Flexural

Isostasia e flexura da litosfera

Isostasia Local

Isostasia Flexural

Isostasia e flexura da litosfera

Isostasia Local

Isostasia Flexural

Carga suportada pela isostasia

Isostasia e flexura da litosfera

Isostasia Local

Isostasia Flexural

Carga suportada pela isostasia

Carga suportada pela isostasia + rigidez

da litosfera

Espessura elástica efetivaEspessura elástica efetiva da litosferaTe :

Te = 0

Te ! 1

Te finite

Espessura elástica efetivaEspessura elástica efetiva da litosferaTe :

Te = 0

Te ! 1

Te finite

Espessura elástica efetivaEspessura elástica efetiva da litosferaTe :

Te = 0

Te ! 1

Te finite

Espessura elástica efetivaEspessura elástica efetiva da litosferaTe :

Te = 0

Te ! 1

Te finiteCarga suportada

apenas pela rigidez da litosfera

Global mapto evaluate the extent to which a region is approachingits final, steady-state, flexure.

Figure 21 shows a 2! 2" global Te map. The map(Figure 21(b)) has been constructed from 18 630individual Te estimates, the distribution of which isshown in Figure 21(a). Because it is difficult, in the

case that Te is high, to precisely constrain the abso-lute value of Te (see, e.g., the discussion in Perez-Gussinye and Watts (2005)) values of Te > 65 kmhave been assigned to 65 km. Figure 21(a) showsthat the data distribution is uneven, especially inthe western Central Atlantic and parts of Africa,

–60 –60

–60–60

0

00

0 5 10 15 20 25 30 35km

40 45 50 55 60 65 70

0

6060

(a)

(b)

60 60

Figure 21 Global Te. (a) Location map of Te estimates (red dots). The estimates are derived from studies of gravity anomalyand topography/bathymetry using both forward and inverse (i.e., spectral) modeling techniques. Principal data sources areEurope, Perez-Gussinye and Watts (2005); Australia, Simons et al. (2003); Africa, Hartley et al. (1996); North America, Bechtelet al. (1990); Lowry and Smith (1994), Armstrong and Watts (2001); Ireland, Armstrong (1997); Former Soviet Union, Koganet al. (1994); India and Tibetan Plateau, Jordan and Watts (2005); South America, Stewart and Watts (1997); and the oceanbasins, Watts et al. (2006). (b) Global Te map. The map is based on a 2! 2

"grid of the Te estimates in (a).

30 An Overview

Te

Watts (2007)

Global mapto evaluate the extent to which a region is approachingits final, steady-state, flexure.

Figure 21 shows a 2! 2" global Te map. The map(Figure 21(b)) has been constructed from 18 630individual Te estimates, the distribution of which isshown in Figure 21(a). Because it is difficult, in the

case that Te is high, to precisely constrain the abso-lute value of Te (see, e.g., the discussion in Perez-Gussinye and Watts (2005)) values of Te > 65 kmhave been assigned to 65 km. Figure 21(a) showsthat the data distribution is uneven, especially inthe western Central Atlantic and parts of Africa,

–60 –60

–60–60

0

00

0 5 10 15 20 25 30 35km

40 45 50 55 60 65 70

0

6060

(a)

(b)

60 60

Figure 21 Global Te. (a) Location map of Te estimates (red dots). The estimates are derived from studies of gravity anomalyand topography/bathymetry using both forward and inverse (i.e., spectral) modeling techniques. Principal data sources areEurope, Perez-Gussinye and Watts (2005); Australia, Simons et al. (2003); Africa, Hartley et al. (1996); North America, Bechtelet al. (1990); Lowry and Smith (1994), Armstrong and Watts (2001); Ireland, Armstrong (1997); Former Soviet Union, Koganet al. (1994); India and Tibetan Plateau, Jordan and Watts (2005); South America, Stewart and Watts (1997); and the oceanbasins, Watts et al. (2006). (b) Global Te map. The map is based on a 2! 2

"grid of the Te estimates in (a).

30 An Overview

to evaluate the extent to which a region is approachingits final, steady-state, flexure.

Figure 21 shows a 2! 2" global Te map. The map(Figure 21(b)) has been constructed from 18 630individual Te estimates, the distribution of which isshown in Figure 21(a). Because it is difficult, in the

case that Te is high, to precisely constrain the abso-lute value of Te (see, e.g., the discussion in Perez-Gussinye and Watts (2005)) values of Te > 65 kmhave been assigned to 65 km. Figure 21(a) showsthat the data distribution is uneven, especially inthe western Central Atlantic and parts of Africa,

–60 –60

–60–60

0

00

0 5 10 15 20 25 30 35km

40 45 50 55 60 65 70

0

6060

(a)

(b)

60 60

Figure 21 Global Te. (a) Location map of Te estimates (red dots). The estimates are derived from studies of gravity anomalyand topography/bathymetry using both forward and inverse (i.e., spectral) modeling techniques. Principal data sources areEurope, Perez-Gussinye and Watts (2005); Australia, Simons et al. (2003); Africa, Hartley et al. (1996); North America, Bechtelet al. (1990); Lowry and Smith (1994), Armstrong and Watts (2001); Ireland, Armstrong (1997); Former Soviet Union, Koganet al. (1994); India and Tibetan Plateau, Jordan and Watts (2005); South America, Stewart and Watts (1997); and the oceanbasins, Watts et al. (2006). (b) Global Te map. The map is based on a 2! 2

"grid of the Te estimates in (a).

30 An Overview

Te

Watts (2007)

Cadeia Havaí-Imperador

Equação de flexura

D =ET 3

e

12(1� ⌫2)

↵ =

4D

(⇢m � ⇢w)g

�1/4

w =V0↵3

8De�x/↵

⇣cos

x

↵+ sin

x

↵

⌘

224 Elasticity and Flexure

Figure 3.30 Half of the theoretical deflection profile for a floating elasticplate supporting a line load.

well-defined arch or forebulge. The half-width of the depression, x0, is givenby

x0 = α tan−1(−1) =3π

4α. (3.133)

The distance from the line load to the maximum amplitude of the forebulge,xb, is obtained by determining where the slope of the profile is zero. Upondifferentiating Equation (3–132) and setting the result to zero

dw

dx= −

2w0

αe−x/α sin

x

α= 0, (3.134)

we find

xb = α sin−1 0 = πα. (3.135)

The height of the forebulge wb is obtained by substituting this value of xb

into Equation (3–132):

wb = −w0e−π = −0.0432w0. (3.136)

The amplitude of the forebulge is quite small compared with the depressionof the lithosphere under the line load.

This analysis for the line load is only approximately valid for the HawaiianIslands, since the island load is distributed over a width of about 150 km.However, the distance from the center of the load to the crest of the arch canbe used to estimate the thickness of the elastic lithosphere if we assume thatit is equal to xb. A representative value of xb for the Hawaiian archipelagois 250 km; with xb = 250 km, Equation (3–135) gives a flexural parameterα = 80 km. For ρm−ρw = 2300 kg m−3 and g = 10 m s−2 Equation (3–127)gives D = 2.4 × 1023 N m. Taking E = 70 GPa and ν = 0.25, we find fromEquation (3–72) that the thickness of the elastic lithosphere is h = 34 km.

Parâmetros do modelo

Te = 103 ate 105 m

⌫ = 0.25

⇢m = 3300 kg/m3

⇢w = 1030 kg/m3

g = 9.8 m/s2

E = 1011 Pa

Fixos Variáveis

V0 = �1011 ate � 1013 N/m

Descobrir o e o . Te V0

Descobrir o e o . Te V0

Descobrir o e o . Te V0

Descobrir o e o . Te V0

Descobrir o e o . Te V0

Exercício

• Dado um perfil flexural da litosfera, criar um programa que determina o Te e oV0 que melhor ajustam os dados observados.